What is a measurement in the Many Worlds Interpretation? I was reviewing the MWI, and couldn't figure what a measurement is in the MWI? Everett claims that split happens when you measure, but what is it for MWI (not for general QM)?
Yes, I know about decoherence, but here I am talking specifically about the split, and how MWI considers the measurement. Also, decoherence is just a part of the problem. I am talking about actual collapse or a split. If an apparatus is of an intermediate size (neither a macroscopic nor a microscopic), how do we handle 'half-splits'? Since MWT is an interpretation and not just a postulate, this should be handled somehow.
This question is not a duplicate of question about general measurements in QM like this one: What constitutes an observation/measurement in QM?
 A: The crucial thing about MWI is, that the measurement is just an ordinary interaction between two quantum systems. An observer or a measurement device is just something, which can "remember" the measured state, i.e. its own state changes depending on the interaction with the measured system. In formulas: Let $\sum a_n |S_n\rangle$ be the state of a system, $|\psi\rangle$ the state of an observer before the measurement and $|\psi_n\rangle$ the state of an observer who measured the state $|S_n\rangle$. The unitary operator $U$ which describes your measurement is therefore
$$
U(|\psi\rangle \otimes |S_n\rangle)=|\psi_n\rangle\otimes|S_n\rangle
$$
Due to linearity, when an observer performs a measurement on the system, your measurement looks like
$$
|\psi\rangle \otimes\sum a_n |S_n\rangle\rightarrow \sum a_n |\psi_n\rangle \otimes |S_n\rangle
$$
which is a superposition of both observer and system state. The "split" is therefore nothing else than an observer being transferred into a superposition state, nothing magical. However, if you choose your states $|S_n\rangle$ to be orthogonal (which they are in all practical applications), so will be your outcoming "superposition components", and therefore there will be no interaction between the superposition components of the observer. Therefore, you can interpret this result as many observers who all measured a different value, i.e. live in many worlds.
One should be aware of the fact, that this description is by far not independent under the choice of basis. However, this problem is solved by results of decoherence theory: Zureck showed, that there are certain states which are more stable than others, these are the classical states, therefore they form a preferred basis for our measurement.
A: 
What [is] a measurement [...] in the MWI? 

A Stern-Gerlach (SG) device has an incoming beam be split by the Schrödinger equation (applied to the actual experimental setup) into two beams, one going left and one going right. This is not the the measurement, this can be undone; and you can tell whether or not has been undone. Specifically, when a series of eigenstates of an orthogonal spin axis B with the same eigenvalue b sent repeatedly into the device over and each being undone and followed by a spin measurement on the axis B gives eigenvalue b each time, just like the first measurement never happened. 
The no cloning theorem says no device can take an arbitrary state in and produce multiple copies of it, but you can make multiple copies of a known state. For instance you can make a bunch of spin up states by sending a beam into a SG device and selecting the good beam and either throwing the other beam away or flipping the spins of the other beam. So we can make as much as we want of a known result.
So when you have an input into a SG device you can't amplify information about that incoming state, it is what it is. But you can set up a metastable system on the two places the split beam will go. Each one can then be used as a detector of sorts, the beam intersecting that region can facilitate interactions that can be spread throughout a large system with many degrees of freedom.
But now if you tried to undo these interactions you have too many things to reverse, it is infeasible to do by accident or on purpose.
So now the original wavefunction has two parts that are no longer going to influence each other (in particular the probability current of the system is the sum of the probability currents of the two, which is generally not true of two states and the current at any place is nonzero for at most one of the branches) so the evolution of each branch is now and forever going to act in all ways as if the other branch is not real.
So far this is still just looking at the Schrödinger equation evolution of the actual experimental setup. In MWI you call these different branches worlds since they act independently and don't affect each other now or ever again. They act like a world onto their own.
Each world has information, the kind that can be amplified, you can call it classical information and the two worlds have different classical information. The information is part of an objective history of that world, but it is not a fact of the full solution that includes all the branches.
Copenhagen tells a similar story, that since each branch acts as if it is the whole world, each branch includes every particle and composite object in the universe. The state of them can be correlated and entangled with other things in that same branch, but they basically act as if other branches (worlds) don't exist.
In Copenhagen you might think the measurement happened at some point, when really it is more like breaking a glass coffee cup them melting it into a new sheet of glass, there isn't really a moment when it becomes impossible to fix the original mug, but at some point it is definitely just too hard to fix it.

Everett claims that split happens when you measure, but what is it for MWI (not for general QM)?

MWI and QM are the same. At some point the wave functions are orthogonal and non overlapping and will (because of hr huge number of degrees of freedom) forever remain that way and when that happens the measurement can be called finished since now the different outcomes act as if the other one isn't a factor.

Yes, I know about decoherence, but here I am talking specifically about the split, and how MWI considers the measurement. 

There is no sharp line, even if they split and are orthogonal, it isn't until that has been spread around to so many degrees of freedom that getting them to overlap again is a nonevent, only then do you have an Everett branch or have distinct worlds.

I am talking about actual collapse or a split. If an apparatus is of an intermediate size (neither a macroscopic nor a microscopic), how do we handle 'half-splits'? 

Its back to the breaking of the coffee mug, maybe when it shattered you could fix it, maybe you could start to heat it a little bit and still hope to catch all the outgoing heat and reverse the effects of everything. But there isn't a line where it becomes truly impossible, it just becomes too hard to happen by accident or on purpose.
Imagine your friend thinks that collapse happens. So they say at first that it happened when the SG device splits the beams. Then you deflect them to get the two. Exams to interfere, thus proving both parts of the original beam were still real.
Then imagine you friend says we need each beam to interact with one thing before the collapse happens. So you do that experiment, and this time you have to undo that interaction in each branch and the original splitting of the incoming beam before you get the interference to show that both results were still there and therefore still real and so no collapse happened.
Then your friend says they need to interact with 2 particles. You show them wrong.
Then your friend says they need to interact with 3 particles. You show them wrong.
Then your friend says they need to interact with 4 particles. You show them wrong.
Then your friend says they need to interact with 5 particles. You show them wrong.
At some point your friend can name a number of particles so large that your current technology can't show they are wrong. That is not evidence that they are right. In fact technology advances and later you do.
No one has ever seen evidence of collapse. To do so would be evidence that the Schrödinger equation is wrong.
The difference between MWI and Copenhagen is solely about the story you tell  when you focus on a particular branch. Copenhagen says the other branches don't exist, MWI says each branch acts as if it were a world unto itself. Only the stories differ, nothing worth making a fuss over.

Since MWT is an interpretation and not just a postulate, this should be handled somehow.

Asking how to handle an intermediate process isn't different for any interpretation. If there is still a way the results can influence each other then do not talk about collapse and do not talk about different worlds, just stick to the full solution as given by the Schrödinger equation.
When you have a split that has progressed to different Everett branches you can renormalize each wave to unit size (and rephase the whole thing) if you feel that is worth your time (the overall phase and magnitude don't have to affect any predictions if you learn how to compute with non unit wavefunctions). When they might still interact with each other you need to keep their relative phase and magnitude. And keep track of exactly how they evolve relative to each other.
This is like when someone asks how many particles you need before you can use statistical physics. At some point it becomes easier to do way more particles and use statistical physics than to track a couple of more particles. But exactly where that cut off is is based on how good you are at dealing with a large (but not statistically large) number of particles.

In my view, what we observe is consistent with Copenhagen and inconsistent with [the Schrödinger] equation.

If what we saw was inconsistent with the Schrödinger equation we would be able to detect that deviation and make different predictions, we don't and that is because there aren't deviations from Schrödinger. But the fact that you think the Schrödinger equation and Copenhagen make different predictions just means you've failed to see that the only thing Copenhagen does is provide a computational shortcut to use a wave for a single instance as a shortcut to compute the relative frequency reading of an aggregator that measures interactions with a large ensemble of identically prepared subsystems.
You can either write the Schrödinger equation for the actual large ensemble of identically prepared subsystems, the devices they interact with and the aggregator that measures the relative frequencies. A method that gives the results we actually see in the lab by accurately describing what you do in the lab. Or you can write just one wave for just one instance of the identically prepared subsystems and use the Born rule. Your choice. You get the same answer in the cases where the Born rule works.

Suppose we have a [SG device] and then, [a] screen. We shoot one electron with some wave function. It leaves EXACTLY one spot on a screen. I understand that this process is undoable

It isn't. You added the detector/amplifier. That causes interactions with so many other things that you aren't going to be able to overlap with the other beam either on purpose or on accident. It's just too hard now.

Schrödinger equation claims (or at least implies) that we will see both spots since both branches exist. 

It does no such thing. This is like when people say that time can't be relative and then claim that special relativity is inconsistent because they didn't want to learn what the theory said.
The Schrödinger equation is quite clear that there is a configuration of the entire universe with that one electron deflected left and a configuration of the entire universe with that one electron deflected right and that the detectors cause the two configurations to evolve into such a chaotic beam in the high dimensional space where that different location interacts with other things that the two configurations never get close enough again for the support of the wave functions to overlap ever again. Which means they each mathematically act as if they were the only one.
So they each act as if a spin up or a spin down electron went through the device. They each act like there is only one electron. You claiming otherwise is just you m is understanding what the Schrödinger equation says. At some point if people tell you what the equation does and says and you keep ignoring that then it is just you choosing to misrepresent the theory. Just solve the actual Schrödinger equation for the actual experimental setup, you don't have to trust me.
It is the whole universe that interacts with the electron the configuration with the electron deflected left starts to have the parts other than the electron evolve differently. That is what is means to see something, it means you act differently and change yourself based on the configuration of the thing you see. Claiming you see something without changing yourself is misleading (there are passive measurements but that requires that you act differently in the other configuration, if you don't change for either configuration you aren't seeing it)

Everett claims that we see just one spot because the world is split when measuring?

The Schrödinger equation says the beam splits. Anyone that says otherwise can be disproven by experiment. Everett says that when two branches have interacted with enough things that they can ignore the other branch because they can't interfere with each other any more because they can't overlap in configuration space. And again the Schrödinger equation says this. Everett just says that means the two branches can have the configurations corresponding to scientists seeing different results can each pretend like the other branch doesn't exist.
Everett says the exact same thing as Copenhagen.

But what happens when we make our screen thinner and smaller?

Did you read my answer? If it is too small then you can't ignore the other branch entirely. But there are certain kinds of things called detectors. When you make a detector you want it to be sensitive to something and not to give different results based on irrelevant stuff other things do. So it needs to have some freedom to be slightly perturbed by other things without changing its gross state and that gross state is the real detector.
Since that is insensitive to small effects from other things it can also sometimes make you insensitive to small effects of overlaps with other branches. But macroscopic quantum things exist, lasers, superconductors, etc.
The Schrödinger equation still describes them. Making incorrect opinions about when you can ignore the Schrödinger equation will just fail you sometimes.
When the screen is thinner it starts to become sensitive to things other than the electron and it starts to become easier to have the two branches still interact with each other.
But that is also what happens in the lab. I'd there something so very very mysterious about being told that you can write down the actual Schrödinger equation for the actual setup (electron, SG, and the screen detector/amplifier)? When you change any one of them you have a new equation to solve, this is not mysterious. You can have no detector and the SG is easy to reverse. If you have a single particle detector becomes harder but still doable, when you have two it is even harder. And so on.
Everything I'm telling you is just using an equation that hasn't been shown wrong and applying it to each situation. People might have told you ways to approximate some of the gross features of the correct model using vast shortcuts of mathematics. That's fine. But it doesn't change anything. And if you go to a situation where your approximation doesn't t hold then go back to the correct equations.
You are like someone that uses finite order multiple expansions instead of finding the actual field and then arguing about when we have to abandon the monopole or dipole approximation and then going so far as to say the actual solution is wrong because you are used to only using a few multipoles. And then you make it sound like this is deep because you thought the dipole was reality.
If you want to know how a medium sized detector behaves, it behaves just like any other detector, but if it doesn't have enough degrees of freedom to be sure it doesn't interact with the other branch then no approximation dirty trick is going to simplify your calculation so you are stuck with the messy reality if a large interacting quantum system.
That's like. If you ask how a medium number of objects interact quantum mechanically then you don't get to simplify. Deal.

Daniel claims that the branches were always there, and no split happened. You claim that branches emerge due to many degrees of freedom (entropy consideration).

No no no. Daniel and I do not disagree. We both agree with the Schrödinger equation as does Copenhagen, MWI, and dBB. We have to. Because we have to agree with the early stages of what happens in an actual SG device. The only place where my answer and Daniel's answer differ is that I am more explicit about telling you where and how the Everett branch forms.
The original split of the position of the electron truly splits the wave into two non overlapping parts. But they are not Everett branches yet. That doesn't happen until those changes in electron position have influenced the configuration of enough other particles that there is no way the two configurations will ever cross again.
There really was a wavefunction with some thickness, and like a river approaching a delta it truly forked, but that fork, that split, is not a fork into Everett branches. That happens when the forks have wiggled to the point where you can't get them to meet again.

But my question is as easy as this: 1. explain why we see one spot, not two on the screen.

The same reason you only see one spot if you removed the SG device but left the electron and screen. I truly can't fathom why you think anything else. Imagine someone asked you to explain why one electron heading towards a screen makes one dot. Whatever you say to them, I'll say the same to you. You had a configuration of the whole universe, which include three degrees of freedom for the position of that electron. As the configuration of all the particles evolves (i.e. as the probability current makes the streamlines evolves) to have the position of that electron go through the device then at first the configuration of the rest of the particles might just continue in an oblivious manner, this is before the Everett branch has formed. But later, for instance when the electron approaches the screen, the configuration of the screen will change in one way if the electron went left and will change in a different way of the electron went right. So the wavefunction has support (roughly, places where it is not zero) in two places, configurations with the electron on the left and the screen reacting to that and configurations with the electron on the right and the screen reacting to that.
There are only three degrees of freedom in the configuration space for the electron and the screen interacts with stuff nearby (you designed it to not be reacting to random other particles from other things). So the screen is designed to only change its configuration if the configuration of all particles in the universe has an electron get near it. And you built your screen to not be detecting random other particles, so there are barriers to other electrons getting near it. There is only one electron in the universe that has a chance to get to the screen. And it can in fact only change parts of the screen and not others because it can't be everywhere. And all of this is true before we stick the SG device between the beam and the screen.
If you stick the SG device in there, it would deflect a spin up beam one way and the detector would be forced to evolve in a particular way in response. Because of how it was designed.
If you stick the SG device in there, it would deflect a spin down beam the other way and the detector would be forced to evolve in a particular (but different) way in response because of how it was designed.
One response for each beam. But different responses. So different configurations of the screen in response to different configurations of the one electron. So different configurations of the electron leads to different configurations of the screen.
Please realize that an electron is 3 degrees of freedom in the configuration space of all particles in the universe. You can have a wave with support on 1,2,3, or a thousand different disjoint regions, but it will still just have one x,y and z in the domain to specify the location of that one electron, that's the difference between configuration space and a function defined on configuration space.
Let([1,0,0],[5,4,0],[-5,4,0]) represent the configuration of the electron (first three) one detector (next three) and the other detector (last three). Over time a spin up beam might have the part of the beam starting at ([1,0,0],[5,4,0],[-5,4,0]) evolve like
([1,0,0],[5,4,0],[-5,4,0])
([2,1,0],[5,4,0],[-5,4,0])
([3,2,0],[5,4,0],[-5,4,0])
([4,3,0],[5,4,0],[-5,4,0])
([5,4,0],[5,4,1],[-5,4,0])
Notice the detector changed its configuration only when the electron got close. Spin down might move like
([-1,0,0],[5,4,0],[-5,4,0])
([-2,1,0],[5,4,0],[-5,4,0])
([-3,2,0],[5,4,0],[-5,4,0])
([-4,3,0],[5,4,0],[-5,4,0])
([-5,4,0],[5,4,0],[-5,4,1])
Each of those configurations evolves to only influence the detector when it gets close. I don't have to make this up, if you write down the Hamiltonian for your actual detectors we can see how they interact with electrons.
Note that there are only 9 components of the configuration. So only one detector reacts.
There is nothing I can do to change this, and you asking me repeatedly to explain why something doesn't happen that the mathematics doesn't allow to happen is like asking me to explain why unicorns don't appear. maybe you had an instructor or TV show or pop science book or something tell you that unicorns appear, but the Schrödinger equation doesn't have room to have unicorns appear, it just has functions defined on configurations. The detectors are designed to only change their configurations when another particle gets close, so when the 3n component vector describing the positions of all particles has one of the particles get really close to the same spot as the detector. So if the configuration of something approaches the configuration of the detector then the detector reacts. You can think of the hyperplane $x_1=x_2$ and the hyperplane $y_1=y_2$ and the hyperplane $z_1=z_3$ and their intersection is a surface of dimension $3(n-1)$ when there are n particles in the universe, only when the configuration approaches that surface does the detector react. When the configuration approaches the surface that is the intersection of the hyperplane $x_1=x_3$ and the hyperplane $y_1=y_3$ and the hyperplane $z_1=z_3$ then the other detector reacts.
There is geometrically no way one electron can get close to both detectors. This is obvious by looking at the configuration space. The only reason I can think you imagine otherwise is because of lazy descriptions of quantum mechanics. Ignore every single lazy description of quantum mechanics, and instead just look at the Schrödinger equation, it is a function defined on configuration space, and it obeys a Partial Differential Equation. This is not deep, this is not complicated.
Each configuration has the electron right next to at most one detector. The wavefunction assigns a complex number to each configuration. But each configuration has the electron near only one detector. I am arguing against a lazy description of quantum mechanics that is not indicated by the Schrödinger equation, because you need me to tell you over and over again to use the Schrödinger equation because you are refusing to give up your baggage. Like people that think the uncertainty principle causes the Schrödinger equation instead of vice versa.
OK, so what really happens. Remember those configurations I listed. The wavefunction is zero in lots of configurations and is nonzero only near those values. You can track the flow of the wavefunction with the so called probability current. Each configuration with nonzero amplitude corresponds to just one detector going off.
Before the Everett branch is formed the wave it has a nonzero value near two very different configurations. Configurations where the electron is approaching one detector and different configurations where the electron is approaching a different detector. But each configuration has just one detector that is about to go off.
What is so bad about a function that is nonzero in two places? Nothing is wrong and nothing is bad. But at this point the two regions of waves don't overlap, but what if later they did overlap, that will affect the probability current. So we still need them both, we have to write the function as something that has nonzero values over two regions. We could even write the function as $\Psi=\Psi_1+\Psi_2$ where $\Psi_1$ is nonzero in one region and $\Psi_2$ is nonzero in a disjoint region.
But after the detector reacts, it will affect other things which affect others and so forth until it becomes impossible to get the evolving $\Psi_1$ and the evolving $\Psi_2$ to ever have the changing regions where they are nonzero to overlap.
But each configuration where $\Psi$ is nonzero corresponds to just one detector going off. And each region where $\Psi_1$ is nonzero corresponds to just detector 1 going off. And each region where $\Psi_2$ is nonzero corresponds to just detector 2 going off. So this is why just one detector is going off. The support of $\Psi_1$ (roughly, the region where it is nonzero) contains configurations for you and me and every other person and those configurations are of us either 1) not caring (going about our business) or 2) are of us interacting with the things that interacted with detector 1 going off or else 3) they are configurations of us interacting with the things that interacted with detector 2 going off. So if you asked how we evolve, there is support (nonzero complex numbers) for configurations with us saying words like "I definitely saw detector 1 going off" and there will be support (nonzero complex numbers) for configurations with us saying words like "I definitely saw detector 2 going off" but there is no support for us saying we saw two detectors go off, since there are no configurations corresponding to both detectors going off that have nonzero complex numbers.
Once $\Psi_1$ and $\Psi_2$ can act without regard for whether it overlaps the other one (because they won't overlap), those physicists saying things like "I definitely saw detector 1 going off" can start using $\Psi_1$ instead of $\Psi$ and it will work just as well. That's all that is gong on. The real wave is $\Psi$ but for those configurations in the support of $\Psi_1$ can use $\Psi_1$ instead.

In my view, it definitely contradicts [the Schrödinger] equation which has both branches.

The Schrödinger equation never assigns nonzero values to configurations where multiple detectors have responded to electrons. Having the view otherwise just tells me you either refuse to setup and solve the Schrödinger equation or you just can't be bothered. The solution to the former is to do it, the solution for the latter is an attitude adjustment and then just doing it.

Use your theory to predict what happens when our screen gets thinner and smaller. 

Eventually it gets so small that you have to keep both branches because interactions can get the waves to overlap. But the answer is simple, you just solve the Schrödinger equation, but not a cartoon version of "it could be here or there or both", just have a function defined on a configuration space and have it evolve according to the Schrödinger equation. Don't expect anything other than the Schrödinger equation to happen. So there will be support for different configurations, but the point is they aren't really different they are overlapping.
Any that's what happens, are there multiple ways to get to the same configuration? If yes, then they can interference and you need to track the phases of those different paths and the Schrödinger equation tells you how to do that, so use it. If no, then don't worry about all those different disjoint regions of different configurations.

Here in my view MWI doesn't say anything which forces me to dismiss this theory as an ad-hoc.

I can't figure out what you are saying, the MWI is telling you when you can ignore different regions of support. Ignoring that can at worst make you wrong and at best (when you happen to guess the right place to ignore it) it can not make you wrong. It's like closing your eyes at an intersection and driving when you think the light is green. If you open your eyes quickly enough after you start moving it doesn't really change anything if you guessed right. And maybe you have good experience guessing and so don't need the detailed information you'd get by actually checking before you go with your eyeballs.
Think of it this way. MWI and dBB and even Copenhagen with a detailed model of the measurement device can tell you exactly when it is OK to simplify your mathematical description. Or you can guess. The first approach will be right. The other one can be right too, but could also be wrong especially if you walk out of the region where historically the physical intuition in your field was honed. The first approach always works, it's called being right.
The MWI is just being right (and so is dBB and so is Copenhagen when you model the actual setup) and that doesn't make you wrong if you do something else, but your rules of thumb is redundant at best. If being right isn't important to you, don't worry about anything I wrote. After all, you might get lucky.
But never think you understand the universe if you can't figure this out. 

Also the fact that forces me to accept collapse is the fact that if my screen is very thin so that an electron leaves a spot but gets out of the screen, and then we put another screen right after the first one, the second spot will be exactly at the same place as the first one.

The Schrödinger equation predicts that too. Please stop thinking the Schrödinger equation predicts something other than what we see in the lab, if anything disagrees with what you see in the lab, it is some lazy version, it isn't solving the actual Schrödinger equation. After the fork (but before the Everett branch forms) each of those waves with disjoint support has the configurations tracked by streamlines that have a so called probability current that is determined by just that wave with support in that region. Remember how the configuration could have that electron be near two detectors? Well that isn't true if one detector is just behind the other one.
Just use the Schrödinger equation, not some hybrid. These two waves with disjoint support each have just the detectors on the left react or have just the detectors on the right go off. Because each wave has support in just the left or just the right so the configuration only gets those detectors close to the electron.
The only reason we can't pretend there is a collapse yet is if those detectors have reacted to the electron but done it in a way that could still be undone. It that is the case we still need to track, because by carefully unreacting the detectors we could get the two waves to overlap in the future even though they don't overlap now.
Not overlapping means they act on their own, there is a reality to the reactions and evolution each does. But that reality can't be described as a simple "either this happened or that happened" since the relative phases and amplitudes of those different waves with currently disjoint support might still be relevant for future predictions about what happens. Not overlapping, and never going to overlap again, that means you have distinct Everett branches.
A: There is no collapse in the Copenhagen Interpretation and there are no "splits" in the MWI. Copenhagen simply says that you are living in one observable universe and that you can't ever fully measure the quantum state of that universe. MWI says that all physically possible worlds exist at the same time, but they don't communicate. So while all possible physical worlds taken together have all the information about their multiverse, they can't put the puzzle back together, which amounts to exactly the same result for every single "slice". 
Philosophically MWI was total nonsense from the outset. It violates Occam's razor on the largest possible scale without getting a dime's worth of physics in return. 
