Problems with many worlds (at great lenght)
What would be some consequences of the many world theory?
Strictly speaking there are no 'consequences' of the many world theory that aren't given by the standard quantum formalism. We simply have to judge whether it is a simpler or more elegant interpretation of QM. Many interpretations contain features that are hard for physicists to swallow. In the case of many worlds, it is the vast multiplicity of seemingly unobservable universes. The interpretation has to provide significant benefits in clarity if we are going to accept this.
Having said this, many-worlds has proven to be useful in that the development of the first interesting quantum algorithm was an attempt by David Deutsch to show that many worlds must be true. We don't need many worlds to understand the algorithm, but this highlights how a diversity of approaches to QM can yield real results.
Why is this theory so hard to believe?
I don't think it is hard to believe; no more so than many of the other interpretations of QM. Some physicists believe that it is obviously the only way to interpret QM, notably David Deutsch. Many in the Quantum Computing community seem to agree on some level, but there is nowhere near universal acceptance amongst physicists as a whole.
I know that slyboy had some points, but can you explain them again? I didn't really understand what was meant.
OK, here is an attempt to explain them as simply as possible.
1) Probability rule: Many worlds says that whenever a quantum decision is made (for simplicity let's say that there are always two possible choices) the universe splits into two universes. For example, one where a particle goes to the left and one where it goes to the right.
However, QM predicts not only the possible alternatives, but also a probability associated to it. For example, there might be a 75% probability of the particle going left and a 25% probability of going right. How do we account for this if there are only two universes?
One way of doing this would be to say there are actually 3 universes where the particle goes left and 1 where it goes right. The probabilities are then given by the relative number of universes corresponding to each choice. This goes beyond the standard many-worlds approach originally proposed by Everett.
The problem with doing this is that the probabilities could conceivably be any number you like, so we might need a continuous infinity of universes for each quantum decision.
Other alternatives for getting quantum probabilities from many-worlds have been proposed, notably a decision theoretical approach by Deutsch. However, these also have their own problems.
2) The basis problem (this one is a bit more technical I'm afraid)
Imagine that we have a detector that tells us whether a particle goes left or right and there is a 50/50 chance of each possibility. One possible wavefunction for this would be:
|particle left>|detector left> + |particle right>|detector right>
Many worlds says that the two terms in this wavefunction represent a splitting across two different universes. However, in QM there is no special significance to this decomposition and we might equally well write:
(|particle left> + |particle right>)(|detector left> + |detector right>) + (|particle left> - |particle right>)(|detector left> - |detector right>)
This could be interpreted in a many worlds fasion by saying that we have two universes each of which has a different superposition for the particle and for the detector.
This may seem arbitrary, but it is not as stupid as it seems. One example that highlights this is the spin states of particles. In this case, (|left> + |right>) is often the same as the state |up>.
Many worlds therefore has to provide an explanation of why one possible splitting occurs and not any other. Attempts to do this are usually based on decoherence, which is to do with the fact that the detector is a macroscopic object and thus its |left> and |right> states couple to the world differently. These explanations are not very convincing to me at the moment.
3) My third problem is not really that serious on a technical level. I'm just saying that we could describe classical physics in the same way as the many-worlds interpretation of QM. The fact is that we don't, because there are no benefits to introducing a multiplicity of universes in this case. Is the phenomenon of quantum interference really any more of a reason to postulate all these extra universes? I don't think so, but it would be a boring world if we all agreed.