# Probability and the many-worlds interpretation

If I toss a coin, then according to the many worlds interpretation of QM, in half those worlds I'll get a head. If I then toss again, then in a quarter I will have got two heads. And so on. There will therefore be some extreme worlds where I always get heads. What happens to the normal distribution of probabilities say in a world where I always get a head, you always toss a six on a die and all electrons are spin up? Similar extreme outcomes of will not happen in only one world, but an infinite number. In these worlds the normal distribution of events will not occur. What an I missing about the many-world interpretation of QM?

• To a large extend, tossing a coin is a classical procedure. If we do not look at the coin nor compute the whole procedure with extremely accurate initial condition, the result in QM language should be $| \mathrm{Side A} \rangle \langle \mathrm{Side A} | + | \mathrm{Side B} \rangle \langle \mathrm{Side B} |$. If we look at the coin or compute the whole procedure with extremely accurate initial condition, we know the result is one side. There is no measurement problem involved. – user26143 Mar 10 '14 at 15:06

Without entering the quantum mechanics of the situation, we can see that each toss is a new world. The next toss is another world, so the series of heads do not add in the way you think to make a world of all heads.

Each world deserted by each new toss will have the usual probabilities of heads or tails.

A world of all heads is possible with sequential tossings making a history of all heads, but not in the way you think:

There will therefore be some extreme worlds where I always get heads

The "always get heads" assumes that you have freedom to keep tossing in the same world. You can only "always have gotten heads" in one world line.

The many worlds interpretation is just mathematics made visual, in my opinion.

Of course to even register that such a world line exists innumerable numbers of worlds will have been created so as to have the history in your world line that such a world existed ! Thinking mathematically is much simpler.

• You can only 'always have gotten heads' in one world line. Wouldn't it be infinitely many world lines? E.g. there's a world where he always got heads and I always got heads, a world where he always got heads and I always got tails, a word where he always got heads and I've never flipped a coin because I lost both of my hands in a freak accident... – user45623 Jun 30 '16 at 1:14

Your observation is correct. If tossing a coin were a quantum measurement, in the many-worlds interpretation of quantum mechanics (MW), there would be a branch of "worlds" in which the outcome was always heads.

This would not violate anything we know about probability or quantum mechanics. The full ensemble of worlds would have the expected binomial distribution of outcomes. e.g. for two coin tosses, there would be one world with $HH$, two with $HT$ and $TH$, and one with $TT$. The probability that "you" live in the $TTT\ldots$ branch is exactly equal to the probbility of there being a single Universe in which you toss a coin and see $TTT\ldots$ You wouldn't be able to infer anything about the Universe/MWI if you saw $TTT\ldots$, other than that you had witnessed something remarkably unlikely.

I will add a few comments regarding MWI and the most important aspect of probability in the MWI. This discussion is all a bit misleading without them.

The "branching of worlds" is an macroscopic phenomena that emerges around the time that a pure state is significantly de-cohered. It cannot be understood from the microscopic theory, from the bottom-up.

In the MWI, there is no collapse of the wavefunction, just unitary evolution of the wavefunction by Schrodinger's equation. The MWI doesn't contain a Born rule for probabilities!, $$P_a = |\langle a | \psi \rangle|^2$$ That's very problematic. The branching occurs around the time a pure state has decohered, such that the basis states don't interefere. There is one branch per basis state. e.g. $$|\psi\rangle = \frac35|\phi\rangle + \frac45|\chi\rangle$$ would not result in 25 Universes, 9 in state $|\phi\rangle$ and 16 in state $|\chi\rangle$, but two branches, one $|\phi\rangle$ and one $|\phi\rangle$. The correct probabilities do not emerge from the branching.

Probability in MWI is an outstanding problem. There have been attempts to derive the Born rule and arguments that it emerges, but I'm not sure whether any solution is widely accepted even in the MWI community.

• Could you have a look at this question. I think your added comments are relevant there and might produce an answer. – orome Dec 16 '14 at 19:14

As far as I am aware, we are only able to speculate about these other "Universes" in the sense that they represent an aggregate of potential (indeterminate) wave functions relating to what events may occur in the time direction known as "future". Or might have occured in the "past" but didn't in this universe. Every time you toss a coin and look at it, that wave function collapses and you are able to determine which universe you are in. Each added toss of the coin adds to the number of collapsed wave functions - and thus potential alternative wave functions which did not occur in this universe.

Reversing time's arrow would seem to decrease the number of alternative possible wave functions, and thus decrease the number of "Universes".

Assuming that you are not an expert coin tosser and can engineer tails every time through skill, then any random distribution of toss sequences can occur, including a lifetime of tails or a normal distribution of tails V's heads, as you say, not forgetting that an infinite run of heads lies within the normal distribution, albeit at a probability approaching zero. You may discover that you live in the world where you decide never to toss a coin again, that mere decision will not collapse the wave function of any possible future toss, as you may change your mind, and take up the old hobby again.

What you are missing is that it is not a viable strategy to get the probability by counting worlds. The only viable way of calculating the probability in MWI is the Born rule. See this paper especially section 9: http://arxiv.org/abs/0906.2718.