The Many-Worlds Interpretation of quantum mechanics (MWI) has become popular in recent years. Among non-physicists, this popularity may be due to the woo factor - imagining a parallel universe in which I won the Nobel Prize, or I didn't eat the last donut, can be lots of fun. However, it has also achieved a certain popularity among physicists - this I find less easy to understand. Let me explain.
In quantum mechanics, we often have to deal with superposition states. These are states in which more than one outcome for some measurement is possible, and we only know the probabilities of each outcome. For example, the state
(|+> + |->)/sqrt(2)
is a superposition state of a spin 1/2 particle in which the "spin-up" state (represented by |+>) and the "spin-down" state (represented by |->) are equally probable. (Here "up" and "down" are defined with respect to some arbitrarily chosen reference direction - not with respect to gravity.)
In the MWI, a superposition state is taken to mean that the world has split into two branches. In one of these branches, the particle really is in the spin-up state, and in the other branch, the particle really is in the spin-down state.
To discuss the MWI it will be helpful to have an alternative interpretation to refer to. My preferred interpretation (at present) is the statistical interpretation (SI). In the SI, any state refers to a collection of identically prepared systems. A superposition state like the one above represents the case where half of the collection is in the up state and the other half is in the down state.
Now, the weird thing about quantum mechanics is that a superposition state is fundamentally different from merely saying there is a 50% probability of spin up and a 50% probability of spin down. That's because a superposition state can exhibit interference, as in the two-slit experiment.
OK, now that we've got some of the basic ideas down I can start to explain what's wrong with the MWI.
Metaphysical excess: In quantum mechanics, superposition states are formed all the time. According to the MWI, that means that the universe is constantly splitting into new branches. So the MWI posits a near-infinity of additional universes, none of which can be detected from whatever branch we happen to land in. This seems to run afoul of Occam's razor.
Compare this to the SI, in which we simply talk about many runs of the experiment. (We need to do many runs in any case, in order to test the probabilities that result from quantum mechanical computations.)
Worlds don't really split in the MWI: We might expect that, once the universe has split into two branches, those branches no longer have anything to do with one another. However, if that were the case, we would never observe quantum interference! In order to retain interference, the MWI must suppose that the different branches of the universe share a ghostly connection that allows interference to occur.
Compare this to the standard interpretation, in which we say that particles sometimes "behave like a wave." For me, it's much easier to imagine a wave moving through real space than a mysterious connection between universes that are separate (but not really).
The basis problem: When do worlds split? If the answer is "whenever there is a superposition", then we have a problem. In quantum mechanics, we can write a state in any basis we like. So in one basis, a given state will have only one term, while in another basis the same state might have two, five, or an infinite number of terms! So how does the universe know how many branches to split into?
The MWI is not so clear on this. The answer is sometimes given as "when a measurement occurs." Then the appropriate basis for branching is taken to be the measurement basis. But how do we decide what sorts of quantum interactions constitute a measurement? Deciding when a measurement has occured is one of the things the MWI was supposed to save us from. But it hasn't.
Even worse is the answer "when decoherence has occurred." Because, if the evolution of the universe is completely quantum mechanical, as the MWI claims, then decoherence is never complete.
Probability problems: If we have a superposition state that has a 50-50 split, then saying the universe branches in two seems to make sense: we have equal probability of ending up in each branch. But what if the probabilities are 2/3 and 1/3? Or 1/sqrt(2) and (1-1/sqrt(2))? The way MWI deals with this is to assume we start with a large number of identically prepared systems, so that the branching can occur according to the usual quantum mechanical probabilities. But this is just the statistical interpretation, sneaking in through the back door!
The bottom line: The MWI doesn't actually solve the problems it is supposed to solve. It doesn't explain interference, it doesn't solve the measurement problem, and it needs to be supplemented with a statistical interpretation in order to reproduce quantum mechanics. So we have paid a huge price in metaphysical multiplication, and have nothing to show for it!
In my opinion, the MWI is an unnecessary and unhelpful load of metaphysical baggage.