Ivan Seeking said:
What kind of following does MWT have these days? Really, I have had a hard time getting anyone to talk about it.
As far as I know, many leading physicists in the fundamental domains in one way or another adhere to it, even if only implicitly. The reason for that is that they seem to adhere to unitarity as a fundamental axiom. As I pointed out a few times, if you can consider things like Hawking does, taking superpositions of spacetimes with a black hole forming and evaporating, and the spacetime where the black hole didn't form, and calculate path integrals (make the terms interfere) over the different possibilities, it seems to me that you take MWI seriously.
All quantum cosmologists seem to take this viewpoint too (the concept of quantum state of the universe has only a meaning in an MWI setting).
Even Penrose says that he thinks that MWI is the only view compatible with unitary quantum theory ; but he thinks that unitary quantum theory will turn out not to be correct, but only a linearized approximation to a theory where gravity will play the role of the non-linear agent that will introduce a genuine objective collapse. Personally, I take that suggestion seriously. I'm indeed not claiming that MWI is ultimately true ! I'm claiming that it is the most natural interpretation of unitary quantum theory.
All the work on decoherence theory (Zeh, Zurek, Joos,...) has only a meaning in the framework of MWI. Indeed, how could you consider the quantum state of the environment and the measurement apparatus otherwise ?
That said, there are of course still many people taking the collapse view seriously. I think that's because they didn't give it enough thought

. Indeed, I know quite some particle physicists who claim that: 1) the quantum state is the physical state ; 2) collapse is something that happens to the quantum state in the measurement apparatus ; 3) locality holds, and ALL laws of physics are to be formulated in a Lorentz-invariant way. This is funny of course, because you cannot formulate collapse in a Lorentz-invariant way! On the other hand, you CAN formulate unitary physics in a Lorentz-invariant way, and that's what they do all the time.
Another viewpoint is that QM is just a mathematical algorithm that just spews out statistics and that we should not worry about what it is supposed to represent. That's easy: you avoid the discussion.
Is this the only interpretation, or are there various approaches with signficant differences between them.
There's a whole class of different views, but it seems always to implicitly accept that an observer is associated to ONE brain state. Actually the view I have exposed is a bit deviant from the majority of MWI versions: in most MWI versions, all the different terms are "equally valid" and you just "happen to be one of them, at random". The problem most of these approaches have is to derive the correct Born rule for the probabilities, but there are several schemes that, with additional (sometimes not realized) assumptions, one can arrive at "effective Born rules", or because the number of states that count are to be distributed in exactly that way, or because some states of small amplitude are not supposed to count, or because of some game theoretical arguments or whatever.
How do we handle conservation of energy with all of these universes popping up? Where does the energy for the copies come from?
Is there a problem with energy conservation in a unitary evolution ? Don't forget that what is called "different worlds" are just the different terms in the wavefunction !
I don't think you are having a problem with writing the state of an electron as a superposition of momentum states, and then wonder where all these momenta come from in the different terms. Is there no momentum conservation ? Do we now have many times the original momentum of the electron because it is in a superposition of momentum states ? That's because you don't ADD them.
A less spectacular name for it is "relative state interpretation". If you go on the quant-ph archive and do a search on that, you'll find tons of papers.
EDIT:
As to your "I have had a hard time getting anyone to talk about it.", I think that is because MWI-ers are a bit affraid to be dumped as lunatics or crackpots by the Copenhagen crowd. It doesn't sit well within the macho physicist culture to "talk philosophy" sometimes. You have to be down-to-earth and hard on maths. I will not hide that it DOES sound outlandish at first. But the more you read about it, and the more you think about it, the more natural it seems to become.