Many-worlds interpretation

Question

Regarding many-worlds interpretation as an alternative explanation to Copenhagen. If we take the generation or possibility of alternative universes as an explanation for the collapse of wavefunction possibilities, wouldn't that mean that every time any wavefunction anywhere collapsed, it would generate an alternative universe, leading nearly instantly to an infinite number of universes being generated every second? It sounds somewhat unwieldy to me.

Answer 1

When someone says they have an interpretation, it means they are not trying to make different predictions, so let's compare the Copenhagen interpretation CI and the many world interpretation MWI, keeping in mind that they are not trying to disagree with each other.

So let's look at CI, what does it predict? It predicts statistical predictions about the relative frequency of results from ensembles of identically prepared systems. How does it do it? It uses complex numbers, state vectors, and linear operators to extract relative frequencies depending on the ensemble, how it is prepared and what is done to it. If the goal is to predict the results of strong measurements, then you have a set of mutually exclusive and exhaustive possibilities (a maximally set of mutually commuting operators, their joint eigenvalues being the possibilities), each of which is assigned a non-negative number according to the rules, which add to 1, the ratios of these numbers being the predicted relative frequency of the particular observations. The CI gives absolutely no story about what happened to produce a particular outcome, but it does say to how to calculate the frequencies and it tells you your predictions. There is nothing wrong with that, it does what it does. Learning it means you are learning the minimum needed to make your predictions (though you might not learn when to compute the frequencies if you weren't told well when a strong measurements happens), and it tells you not to read too much into other stories about what is happening because two stories that make the same predictions don't really have an edge over each other. Now there are other reasons to consider studying one theory or another besides a desire to make a prediction, maybe some theories are easier to remember, easier to teach, or even easier to modify into new theories or just to inspire you. But be fair to all theories and be honest about why they are succeeding or failing in what you expect of them or hope from them.

Now let's look at MWI. Its goals are to make the same predictions, that what the I in MWI is for. Generally, you make your calculations in the same way, but you say that the reason that a particular result is observed in a strong measurement is that you can image the entire set of possibilities available but that a possibility of one result and a possibility of another result have evolved to the point where they can't interfere with each other in any practical or feasible way. So each of those possibilities can, from its own perspective, pretend that the other possibilities are now "different" universes, and they can basically ignore each other. The point is that MWI doesn't say that just one result happened, all the results happened, but each acts like a world unto its own and so can ignore the other results. The MWI would say that each world sees only its own result and that is just because it can't interact with those other world that got different results. The predictions about the probability that you got one result based on your initial setup is the exact same as the CI, because the MWI was designed to have the same relative frequencies as the CI.

To be fair, the CI doesn't have a collapse postulate any more. The CI now uses decoherence (in the exact same way that the MWI describes branching) to describe when and how a measurement has occurred. The collapse postulate is only a summary of the net effect of what happened from decoherence that you can use after decoherence already occurred to summarize the net effect for a particular result. If you didn't have decoherence, and had a literal collapse, then you'd have to say when it happens (too frequently and you'd get a quantum Zeno effect that was so strong that nothing would evolve, yet have it happen too infrequently and it wouldn't happen enough to explain what we do see when you actually do measurements, so you'd have to have decoherence and branching do the heavy lifting most of the time). So we do have another means besides a collapse postulate, we have decoherence, and it says that the quantum rules for generating probabilities reduce to something very very close to the regular probability rules where you can have non-negative results assigned to exhaustive mutually exclusive results that all add up to one, and further that each of those results can then forever be treated like a normal either-this-or-that probability that already happened, because future experiments will compute probabilities in a way consistent with that division.

It think this is easier with a simple example. If you had a two state system such as the spin of an electron, then you can keep track of the state with two complex numbers, so think of it as a vector $(a,b)$ where $a$ and $b$ are complex. Now you can compute how the vector $(a,0)$ evolves (into another vector of the same length let's say it evolves into $(a/\sqrt{2},a/\sqrt{2})$, and you can compute how the vector $(0,b)$ evolves, say it evolves into $(b/\sqrt{2},-b/\sqrt{2})$ and the evolution of the the vector $(a,b)$ will be the sum of the evolutions of the two $(a/\sqrt{2}+b/\sqrt{2},a/\sqrt{2}-b/\sqrt{2})$. So that's nice. And you can compute the frequency that if you stick it in a particular device that it gets deflected left (take the first component and square it) or gets defected right (take the first component and square it). If so, the first state $(a,0)$ evolves into one that goes left or right with 50-50 chance. And so does the second one. If we assumed that it started out either in the state $(a,0)$ or in the state $(0,b)$ then since either one has a 50-50 chance of going left or right, then whichever one we have will go left-right with 50-50 chance. But what if we don't have it starting in state $(a,0)$ or in state $(0,b)$ and instead it starts in state $(a,b)$. So what if it started in state $(1/\sqrt{2},1/\sqrt{2})$, then we'd indeed predict (and every interpretation/predicts) that it evolves to the state $(1,0)$ so it goes left with 100% probability. So a state can be written as a sum of two states, but the probability that the superposition (the sum of the states) gives a particular result is not in general the sum of the probabilities, or even a weighted sum. But sometimes they can. That happens when certain states are orthogonal, and decoherence says in which situations the end result states are mostly orthogonal and which states are almost entirely orthogonal. Orthogonality matters because the probability is the squared length of a vector and for orthogonal vectors the squared length of a sum is the sum of the squared length, so if they will forever more be orthogonal, then we can computes the rates of one and the rates of the other and get the rates of the sum to be the sum of the rates. And in those situation we can start computing the probabilities and using them according to the normal rules of probability.

So the CI says when you can get probabilities, and says that you can get them when the normal rules for using those probabilities later on will reproduce the results that CI would predict if you waited to use the CI way of getting probabilities. Because the CI only cares about telling you how to get your probabilities.

Whereas, the MWI says that at the same point where CI says that you can start computing some probabilities because it won't affect your answers, they tell a story that different "worlds" have formed and won't affect each other any more so you can find out the chance you ended up in a particular world.

They are both saying that you are free to compute a probability and then act as if one for those things happened, because they both agree that using regular probability on those events will now give you the same results about future probabilities as they wanted to predict. The CI just says that you may compute probabilities at that point because the CI just wants to tell you how to compute probabilities. The MWI wants to tell a story about why you can do that.

Do you need a story to compute a probability? No. So the MWI isn't better at making predictions. Some people think they make different predictions in the presence of a time machine, the whole reason we can use regular probabilities at that point is because the different results aren't going to affect each other, if a time machine allowed them to affect each other, then the assumption about it being OK to compute the probabilities now is wrong and then the CI and the MWI would both agree on saying that we have to keep it as one superposition state, and can't treat it as two states with a probability of being in one or the other.

But this does point at a sense in which the MWI seems different. And that is that the equations of QM are time reversible, so while in practice is might be very very hard to take a decoherent wave and bring it back together, we can't say for sure, just like all the gas might end up in a small portion of a container. The MWI tells a story that makes it seem impossible, when really it is just very very unlikely. But that makes the MWI as wrong as the second law of thermodynamics. And a CI use would probably make the same mistake unless they really took pains to avoid making the same error. And really this is the exact t ambiguity CI has, where it's not totally telling you when it's OK to compute a probability. So it's not really a shortcoming of MWI itself but just a practical simplification that is highly justified in almost all situations.

Now, I'd like to address your question. What's unweilding is keeping track of the amplitude for all these measurements type events that happened in the past when their results are so very close to orthogonal to each other and thus keeping them around just creates such super tiny corrections to what the regular rules of probability will give you. The point of the MWI is to then ignore those other worlds. And even with the CI people do that too but they just use the exact same decoherence criteria, and simply don't say why they do this mathematical simplification other than that it gives almost exactly the correct probabilities. There is nothing unwieldly about either interpretation's dealing with decoherence, they tell you what probabilities to compute, roughly when to compute them, and definitely how to compute them. And the MWI tells a story too. But the story is really just a description of when you can compute those probabilities with enough words in the story to get a sense of why.

Answer 2

It is my understanding that MWI says there is no wave function collapse. The electron is in all the places at the same time, and when there is measurement to find the electron, we are catching one of the places that the electron can be, but in other world's the electron is in point X, but we caught the electron in point Y.

So in other words, I wore green shirt today, but in a different world, I wore a blue shirt. There is almost infinite number of combinations that these things can happen, IMHO.

I personally think, MWI is hard to believe. I think it is an easy way out.

Answer 3

If we take the generation or possibility of alternative universes as an explanation for the collapse of wavefunction possibilities, wouldn't that mean that every time any wavefunction anywhere collapsed, it would generate an alternative universe, leading nearly instantly to an infinite number of universes being generated every second? It sounds somewhat unwieldy to me.

In the MWI the wave function doesn't collapse. Rather, some interactions copy information about the value of an observable from one system to another. When that observable is unsharp, there are multiple distinct versions of the system that records the information where there was only one before. Now, if you look at the state vector before the copying of the information it has a square amplitude of 1 and the same is true after the measurement. So it makes more sense to say that before the copying there were many instances of the copying system that all had the same value for every measurable quantity. After the copying that set has been partitioned into sets that have different recorded values of the copied observable. So in the MWI pre-existing instances of systems are partitioned into sets. And in each of those sets, energy, angular momentum, etc. are conserved, so there is no problem with conservation laws.

You say this is unwieldy, but it is just a consequence of quantum mechanics without a collapse postulate. I don't follow what the problem is supposed to be. Perhaps you could explain your problem more fully in comments.

Now, above some people noted that there are so-called continuous observables and say that this means a continuous number of different universes is generated every second as a result. This is a mistake. There is a continuous infinity of different instances of any system as I explained above, but it doesn't follow from this that there is a continuous infinity of distinguishable versions of a system. Any physically realisable measurement process can only distinguish finitely many values of those continuous observables. For example, if you have an electron microscope, its resolution is limited by the energy of the electrons it produces. If a structure has features smaller than the resolution of the microscope, the microscope won't have different states as a result of interacting with those different features with the electron beam. So the only quantities that can actually be measured all have finite sets of values.

Answer 4

IMHO the MWI is the only "this is what REALLY happens" explanation for non interactive imaging eg an extension of the "bomb tester" where the state of an object is determined with no interaction of the measuring apparatus detectable from the POV of the object under study