And which has more evidence?
What do you mean "which theory" of quantum mechanics is correct? Quantum mechanics is the theoretical framework which has been observationally confirmed quite comprehensively. I think you mean to ask which interpretation of quantum mechanics is the correct one. The answer to that is that no one interpretation has been confirmed as the correct one, as they all yield identical predictions that are fully compatible with quantum theory. Though several have some hope of one day becoming uniquely falsifiable with the advent of new experimental technology, as of now they are emprically equivalent, and thus we can't distinguish between them and determine which one is "correct," if any of them are.
Also, while no one theory is more correct than the rest, each theory has different amounts of people who adhere to that interpretation.
The Copenhagen interpretation is usually considered the most popular among physicists.
The Many Worlds Interpretation is usually considered second most popular, as far as I'm aware.
Other than that I can't say for certain how popular the rest are compared to Copenhagen.
The main point is still that all the interpretations offer the same results. They only explain them in different ways. Until one is able to explain or predict phenomena that the others don't, they will all be considered equally valid.
is the many worlds the only interpretation that assumes parallel universes?
Yes, but there are other ideas about how parallel universes could exist. You could try to find the stuff that Max Tegmark has written about it if you'd like to know more. The article he published in Scientific American with the title "Parallel universes" was a very interesting read.
As far as I know MWI is the only interpretation that makes the positive statement that other universes exist. Not to get into too much detatil, MWI claims their existence, and consistent histories is agnostic about the reality of the decohered branches of the wavefunction. Bohmian mechanics, for examples, denies the reality of any branch but our own. However, again I believe that MWI is the only interpretation that comes out and says these other branches have ontological reality. Other scientific theories do posit the existence of parallel universes however, such as Linde's bubble theory, where our universe sprouted from a small section of an inflation field. A couple of black hole theories also propose their existence. The proposal of the multiverse does run into the scientific problem of being observationally unverifiable at the moment, and this a continuing source of tension in the scientific community...it is an exilarating idea, even if frustratingly unfalsifiable concept.
so MWI is the only one that states parallel universe's unlike the rest?
Yes, MWI is the only interpretation that explicitly states there are parallel universes.
MWI is a little more subtle. It was originally called "relative state", and it said that each branch of the wavefunction continues to evolve unitarily - all the branches add up to 1 - and that there is no "collapse". The state of a particle is relative to the branch of the wavefunction you're in. It doesn't - I don't think - say the other branches *exist* so much as it says "we don't assume they are thrown out." And, further, it is shown that the branches can never interact again in the future - so whether they exist becomes a question that can never be answered.
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