How does many worlds interpretation work for non-50/50 probabilities?

Question

The many-worlds interpretation of quantum mechanics has always been explained to me at a high level using examples of binary events (e.g. atom either did or did not decay at any given moment in time), which leads to a conceptually clean idea of "branching" into two distinct universes. But how does branching work if you have, say, a 70% probability of something happening (e.g. measuring the spin of an electron having gone through an SG apparatus oriented at an arbitrary angle)?

Do you say that 7 universes got to spin up and 3 that got spin down (to account for the 70% probability)? Does that mean those 7 universes are in-every-way identical copies of each other? But then what if you had something with a 71.87154819...% chance of happening? You would need an uncountably infinite number of branches to be able to represent arbitrarily precise probability ratios, and then subsets of those branches would contain uncountably infinite universes that are 100% degenerate and identical to each other. Is this what the standard many worlds interpretation assumes?

If not this, then what? You can't say that there's a 70% chance of universe A happening and a 30% chance of universe B happening if you're saying both happen. How does the many-worlds interpretation put a "weighting function" on different branches or outcomes?

Answer 1

Binary branching is just a simplification to make it easier to explain without math. The actual math is very simple, and can handle unequal probabilities.

At the simplest level, a branching occurs when you can write the wavefunction as a sum $$|\psi \rangle = |\psi_1 \rangle + |\psi_2 \rangle$$ where $|\psi_1 \rangle$ and $|\psi_2 \rangle$ are orthogonal and decohered, i.e. that there is no reasonable physical process that can make them overlap again. In this case we colloquially describe the two terms as "worlds" or "branches", and the probability of being in each one is the norm $\langle \psi_i | \psi_i \rangle$, which can be an arbitrary number between zero and one. The same logic goes for branching into more than two "worlds" at once, and repeated branching: you just get a sum of many terms, and the probability of each one is its norm.

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After some comments, I get the feeling you really want a discussion of where the probability in the many worlds interpretation "comes from". Again, this is a very subjective and debatable thing, but my favorite take on it is "self-locating uncertainty".

Suppose that somebody kidnaps you, blindfolds you, and takes you somewhere in Uzbekistan. When you come to your senses, are you closer to Samarkand than Tashkent? You don't know for sure, so you can only answer in terms of probabilities. This is self-locating uncertainty: you're certainly in a definite place, and it's not like there are many copies of you running around, but there's probability nonetheless. You can use a variety of information to help. For example, if you weight by area, about 85% of the country is closer to Samarkand. (But this doesn't mean there are $85$ copies of you near Samarkand and $15$ copies of you near Tashkent!) But if you weight by population, substantially more of the population is closer to Tashkent, because it's the capital. Of course, which weighting is the correct choice depends on how the kidnappers set things up.

Now, suppose that after the spin of a particle is measured by a device, the state is $$|\psi \rangle = \sqrt{0.85} |\text{spin up measured} \rangle + \sqrt{0.15} |\text{spin down measured} \rangle.$$ You are living in one and only one branch of the wavefunction, but until you look at what the device is reading, you don't know which. At best, you can assign probabilities. The core assumption of many worlds is that the correct choice of probability (i.e. the choice that corresponds to what you actually observe, when averaged over many measurements) is to take the coefficient of each branch and take its norm squared, i.e. to assign an 85% chance to observing spin up.

If you ask where this assumption comes from, it's a perfectly legitimate question! However, the point is, there's no principle that says the probabilities have to be equal across branches. That's like saying every day must have a 50% chance of rain because it can either be rainy or not.

Answer 2

You would need an uncountably infinite number of branches to be able to represent arbitrarily precise probability ratios, and then subsets of those branches would contain uncountably infinite universes that are 100% degenerate and identical to each other. 

What's the problem with having an infinite number of branches? David Deutsch, a leading modern proponent of the Many Worlds interpretation, proposes that scenario in his popular book, The Fabric of Reality. In this picture, the universe started with an infinite number of parallel branches or strands, and at each quantum decision various subsets of those strands diverge, with all of the strands in any given bunch being 100% identical.

This version of MWI gets around an objection that many people have of MWI: at each quantum decision it seems that a whole new universe (or many new universes) needs to be created for the new branch (or branches), and that sounds like a flagrant defiance of the conservation of energy. Deutsch's scheme shifts that problem to the moment of initial creation at the Big Bang.

Personally, I'm not a huge fan of the Many Worlds interpretation, but Deutsch's version is my favourite flavour of MWI. To paraphrase Niels Bohr, it's a crazy theory, but I'm not sure if it's crazy enough to be true. ;)

Answer 3

I am no expert on the many-worlds interpretation, but I always thought that if the wavefunction is $\sqrt{7/10}\psi_1+\sqrt{3/10}\psi_2$ then there is (in this interpretation) one universe in which the wavefunction collapses to $\psi_1$, one universe in which the wave function collapses to $\psi_2$, and you have a 70% chance of branching into the former and a 30% chance of branching into the latter.

I don't think the Born probabilities have anything to do with the number of branches. There is no notion that branching into each branch is equally likely so you have to have 7 $\psi_1$ branches and 3 $\psi_2$ branches.

Instead, the number of branches is simply the number of possible outcomes for a measurement of the observable, which in this case is 2. For a more complicated superposition it could be more, but the number of branches is still unrelated to the branching probability.

Answer 4

Well, first of all, "branching" into different "universes" is a simplification, but I'll put that aside as that's a rather difficult area to explain. A gross simplification is that whenever an experiment can have two different results, each result exists, and the degree to which each exists is proportional to their probability.

So, let's say you have an experiment in which the Copenhagen interpretation says that you have a 70% chance of measuring a particle as spin up, and 30% chance of measuring spin down.

What MWI says is that the initial state can be decomposed into a quantum state in which the particle is spin up, and a quantum state in which the particle is spin down. As the states evolve, they interact with the rest of the universe, and so the states encompass the entire universe, including the experimenter. The first state ends up describing a universe in which the experimenter observes a spin up participle, and the second one a universe where the experimenter observes a spin down particle.

MWI also says that the number 0.7 is associated to the first state, and to the second, 0.3. These numbers are sometimes referred to as "measures", "weights", or "probabilities". Now, where things get a bit fuzzy is what these numbers "actually" are. The simple answer is that they are "probabilities"; the first number says that the "probability" of finding yourself in the first state in 70%. But that just raises the question of what "probability" means. There's no physical "thing" that the word "probability" refers to. One can say that it refers to long term trends; if we had 1000 experiments with "essentially the same" setup (whatever that means), then we should expect to find ourselves observing spin up around 700. More precisely, observing spin up 700 times is the most likely result. But that just defines it in terms of probability again.

Classical physics says "If you do $X$, then $Y$ will happen". Copenhagen interpretation says "If you do $X$, then $Y_1$ will happen with probability $p_1$, and $Y_2$ will happen with probability $p_2$". MWI says "If you $X$, then there's this quantity 'measure' that has a value of $p_1$ for $Y_1$ and $p_2$ for $Y_2$".

Answer 5

The question of probabilities in MWI has been a big problem and several attempts that don't go all the way have been put forward.

The many-worlds interpretation is to my mind truly convincing so that it appears beautiful. It relies only on that the quantum equations that we know describe so many physical systems very accurately also can describe what happens in a measurement. It is the most natural assumption that the detectors also works according to quantum mechanics also when it is used in an experiment. Note, that the Copenhagen interpretation refuses to think of the detector a quantum mechanical system when it performs a measurement, but it is fine to think of it in quantum mechanical terms when its workings is the object of study.

There is actually no creation of new worlds, but a branching of the wavefunction due to decoherence. This branching of the wavefunction is a process that is fully understood as quantum mechanical process involving the system being measured the many degrees of freedom of the detector and its environment which causes decoherence. There is nothing magical, no more of a problem with energy or mass conservation than there is when a wave goes through a double slit. No sane experimenter would ever use as a detector an apparatus as a detector if it doesn't produce decoherence. This is not only self-evident but it is also an empirical fact.

Finally, to answer the question about probabilities. As is well-known from the Born rule, it has to involve the square of the amplitude, but how does that come in. To answer that we have to know the physical significance of the wavefunction amplitude. What does it signify? This was long a neglected question. Everyone took this for granted that "we knew" what it meant. The MWI has abandoned the usual postulates as fundamental assumptions which means abandoning the idea that the amplitude is a probability amplitude as implied by the postulated Born rule. So what is the amplitude in MWI, or rather what is the physical significance of the wavefunction absolute squared? One quantity a mechanical description of the physical world has to deal with is position. What is there to say about position when all there is to describe the physics is the wavefunction. What does it say about position? Well, when the whole of a wave packet is inside some volume, e.g. a detector. Then we will not hesitate to say that the particle is inside that volume (detector). Clearly, the wavefunction contains information about where things are. But there is no point thing in MWI, there is only the wavefunction, or rather what the wavefunction accurately depicts. The answer here is that the wavefunction absolute squared gives where the system is located. THIS IS A ALMOST SHOCKING STATEMENT: The answer to the question where something is located is a distribution given by the wavefunction absolute squared. This is to change the meaning of the location of an object as an electron. Its position is a distribution, not as a probability of where the point location is, because there is no such point thing in this description. There is only the wavefunction which is a distributed object, there is nothing else, absolutely nothing.

What we have, and the only thing we have is this new concept of a location which fundamentally is a distribution. This will give us probabilities in the sense of what we will expect to see when a measurement is repeated many times, because the wavefunction of the system of many identical experiments, when viewed as a function of the relative frequencies will be peaked at the relative frequencies given by the Born rule. That is, the typical location, and thus where we expect to find in terms of relative frequencies the system of many repeated measurements is at the Born rule values. This is in effect what we have the Born rule probabilities for. We have them to make forecasts of what relative frequencies will we find.

For more details on this see http://arxiv.org/abs/1902.05521